Isolated types in DLO

Question

Let $\mathcal{M}$ be a dense linear order without endpoints and $A\subseteq M$. Marker shows that a type $p\in S_1^\mathcal{M}(A)$ which is not realised in $A$ is isolated iff the cut $(L,U)$ over $A$ associated to $p$ is such that $L$ is either empty or has a greatest element and $U$ is either empty or has a smallest element. Is there an easy example (easy = if possible, involving only $\mathbb{R}$ or $\mathbb{Q}$ with their natural order, and not some ordinal...) of such a structure and such a type? I don't have any example in mind, easy or not, so I'll settle for a non-easy example if necessary!

Answer 1

There are very easy examples. Just because $M$ is a dense linear order doesn't mean that $A\subseteq M$ is densely ordered.

Take $M = \mathbb{Q}$ and $A = ([0,1]\cap \mathbb{Q})\cup ([2,3]\cap \mathbb{Q})$. Then besides the types realized in $A$, there are three isolated types in $S^M_1(A)$. They correspond to the cuts $(\emptyset,A)$, $([0,1]\cap \mathbb{Q},[2,3]\cap \mathbb{Q})$, and $(A,\emptyset)$ and are isolated by the formulas $x < 0$, $1<x \land x < 2$, and $3<x$, respectively.

Alternatively, if $A$ is any finite subset of a dense linear order, then every type over $A$ is isolated. This fact is true more generally for any model of any countably categorical theory (by Ryll-Nardzewski and the fact that the theory remains countably categorical after naming finitely many parameters by constants).